Question

\( R \) is a relation on a set of natural numbers \( N \) defined by \[ R = \{(a, b): a, b \in N \text{ and } a = b^2 \} \] Is \( (a, b) \in R \), \( (b, c) \in R \Rightarrow (a, c) \in R \) true? Justify it by one example.

Hint:

A relation is transitive if \( (a,b) \) and \( (b,c) \) imply \( (a,c) \). Check by examples.

Answer 1

Step 1: Check transitivity. If \( (a, b) \in R \) and \( (b, c) \in R \), then \( a = b^2 \) and \( b = c^2 \). 

Step 2: Check if \( (a, c) \in R \). \[ a = (c^2)^2 = c^4 \] Since \( a \neq c^2 \), transitivity does not hold. Example: Take \( a = 16, b = 4, c = 2 \). \[ (16,4) \in R \text{ and } (4,2) \in R \] But \( (16,2) \notin R \), hence not transitive.